Lemma

The number of permutations of with k descents is equal to the number with k drops, i.e.,

Proof

A descent of must lie inside a cycle of since our conventions guarantee that the last element in a cycle is followed by a larger integer. By the meaning of the cycle decomposition (namely, that elements within cycles are mapped to the next element in the cycle) we see that a descent of corresponds to a drop of . Conversely, a drop in must occur within a cycle (i.e., not in passing from the last element of a cycle to the first) and corresponds to a descent in . Thus the number of permutations with k descents is equal to the number with k drops.